| 摘要 | 第4-6页 |
| abstract | 第6-7页 |
| 1 Introduction | 第11-25页 |
| 1.1 CHY Formalism and Scattering Equations | 第13-18页 |
| 1.1.1 CHY forms at tree and loop levels | 第14-16页 |
| 1.1.2 Polynomial scattering equations | 第16-18页 |
| 1.2 Bern-Carrasco-Johansson Relations and Color-Kinematics Duality | 第18-25页 |
| 1.2.1 BCJ numerators from CHY forms | 第20-22页 |
| 1.2.2 Ambiguities of BCJ Numerators | 第22-25页 |
| 2 A Differential Operator for Multidimensional Residue 162.1 Conjecture for The Differential Operator | 第25-43页 |
| 2.1 Conjecture for The Differential Operator | 第26-27页 |
| 2.2 Direct Evaluation of One-Loop Generalization of CHY-Form | 第27-36页 |
| 2.2.1 Residues at finite poles | 第29-32页 |
| 2.2.2 Residues at infinity | 第32-34页 |
| 2.2.3 Residues at spurious poles | 第34-36页 |
| 2.3 Application: Four-Point One-Loop SYM Integrand | 第36-43页 |
| 2.3.1 Computing the residues at finite poles | 第37-39页 |
| 2.3.2 Computing the residues at infinity | 第39页 |
| 2.3.3 Computing the residues at spurious poles | 第39-41页 |
| 2.3.4 Summary of the four-point integrand | 第41-43页 |
| 3 Combinatoric Properties of Polynomial Scattering Equations | 第43-67页 |
| 3.1 Prescription for Determining The Differential Operators | 第44-53页 |
| 3.1.1 Preliminary: notations and tableaux | 第44-47页 |
| 3.1.2 Coefficient generating algorithm | 第47-53页 |
| 3.2 Application: Five-Point One-Loop SYM Integrand | 第53-64页 |
| 3.2.1 Simplifying the one-loop generalized CHY integrand | 第54-56页 |
| 3.2.2 Transformations to the prepared CHY integrand | 第56-58页 |
| 3.2.3 Evaluating the prepared form of the five point integrand | 第58-62页 |
| 3.2.4 An analysis for the spurious poles | 第62-64页 |
| 3.3 Application: One-Loop n-gon Amplitude | 第64-67页 |
| 4 A Reduction Method Based on Differential Operator | 第67-91页 |
| 4.1 A Basis of BCJ Numerators from The Differential Operator | 第68-79页 |
| 4.1.1 Canonical coefficients in the differential operator | 第70-72页 |
| 4.1.2 Discussion on dimension of minimal basis | 第72-75页 |
| 4.1.3 Reduction matrix | 第75-79页 |
| 4.2 Application: Gauge Invariant BCJ Numerators of Yang-Mills | 第79-91页 |
| 4.2.1 Yang-Mills four-point amplitude | 第80-83页 |
| 4.2.2 Yang-Mills five-point amplitude | 第83-91页 |
| 5 Conclusion and Outlooks | 第91-95页 |
| A Appendices | 第95-113页 |
| A.1 Numerical Checks for the Ansatz | 第95-100页 |
| A.1.1 Homogeneous ideals | 第96-99页 |
| A.1.2 Inhomogeneous ideals | 第99-100页 |
| A.2 Standard Method for Dealing With Poles at Infinity | 第100-103页 |
| A.3 Obtaining Residue by Inspection | 第103-106页 |
| A.4 5-Point Complete Result | 第106-108页 |
| A.5 6-Point Reduction Matrix | 第108-113页 |
| Bibliography | 第113-123页 |
| Publication List | 第123-124页 |
| Acknowledgements | 第124-125页 |
